Q22 of 45 Page 1

Find the equation of the tangent to the curve y = root 3x-2which is parallel to the line 4x - 2y + 5 = 0.

OR


Find the intervals in which the function f given by f (x) = x^3 + 1/x^3 , x not equal 0is


(i) increasing


(ii) decreasing.

We know, derivative of a function at any point gives the slope of tangent at that point.


Now,


[1]



Also,


Tangent is parallel to the line 4x – 2y + 5 = 0


Slope of this line is


⇒ slope of tangent is 2


And therefore we have,




⇒ 48x – 32 = 9


⇒ 48x = 41



Putting this in [1], we get





Hence, point passing through tangent is


Therefore, equation of tangent is


y – y1 = m(x – x1)


where, m is slope of tangent and (x1, y1) is a given point on tangent



⇒ 48x – 24y – 23 = 0 is required equation of the tangent!


OR


We know, function is increasing if


f’(x) > 0 and decreasing if f’(x) < 0


Now,





[using a3 – b3 = (a – b)(a2 + ab + b2)]


Now, x4> 0 and x4 + x2 + 1 > 0 [for each x ∈ R ]


So, sign of f’(x) will depend on the (x2 – 1)


Now, x2 – 1 = (x – 1)(x + 1)


Which is zero at, -1 and 1


Now,



Therefore,


f’(x) > 0, if x ∈ (-∞, -1) ∪ (1, ∞)


and


f’(x) < 0 if x ∈ (-1, 1)/{0} [since f(x) is not defined at 0]


Hence,


f(x) is increasing at (-∞, -1) ∪ (1, ∞) and decreasing at (-1, 1).


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